Since \(\partial M\)  is a smooth hyper-surface of \(N\)  there exists a smooth function \(s\colon N \rightarrow {\bf R}\)  and a neighborhood \(U\)  of \(\partial M\)  such that
\(|s(x)|=\text{dist}(x,\partial M):=\inf \lbrace d_N(x,z): \: z \in \partial M\rbrace , \quad \text{and} \quad \Vert \text{grad}\; s(x)\Vert _g\equiv 1,\) 
for every \(x \in U\) . Moreover for each \(x \in U\)  there exists a unique \(z \in \partial M\)  such that
\(d_N(x,z)=|s(x)|.\) 
We choose the sign convention of \(s\)  such that \(s(x)\ge 0\)  for \(x \in U \setminus M\) . Then on \(\partial M\)  the gradient of the function \(s(\cdot )\)  agrees with the outward pointing unit normal vector field of \(\partial M\) . The existence of this function is explained for instance in [1]}.
By this construction, each \(p\in U\)  can be written uniquely as
\(p=(z(p),s(p))\in \partial M \times {\bf R},\) 
where \(z(p)\)  is the closest point of \(\partial M\)  to \(p\) . Thus on \(U\)  we write the Riemannian metric as a function of \((z,\varepsilon )\in \partial M \times {\bf R}\)  in the form
\(\mathrm {ds}^2+\tilde{g}(\varepsilon ,z),\) 
where \(\tilde{g}(\varepsilon ,z)\)  is the first fundamental form of the smooth hyper-surface
\(\Omega (\varepsilon ):=s^{-1}\lbrace \varepsilon \rbrace .\) 
By [1]} we can then write the second fundamental form of \(\Omega (\varepsilon )\) 
as a bi-linear form
\(\Pi _{(z,\varepsilon )}(X,Y)=-\frac{1}{2}\frac{\partial }{\partial \varepsilon }\tilde{g}_{\alpha \beta }(\varepsilon ,z) X^\alpha Y^\beta \in {\bf R}\) 
on \(T\Omega (s)\) . Thus the eigenvalues \(\lambda _1(z,\varepsilon ), \ldots , \lambda _{n-1}(z,\varepsilon )\)  of \(\Pi _{(z,\varepsilon )}\)  are continuous functions of \((z,\varepsilon )\)  [3]}. Since \(\Omega (0)\)  coincides with \(\partial M\) , which is strictly convex, we have that \(\lambda _{\alpha }(z,0)<0\)  for every \(\alpha \in \lbrace 1,\ldots ,n-1\rbrace \) . Thus there exists \(\varepsilon _0>0\)  so that
\(\lambda _{\alpha }(z,\varepsilon )<0, \quad \text{for every $\alpha \in \lbrace 1,\ldots ,n-1\rbrace $ and $|\varepsilon |<\varepsilon _0$}.\) 
Therefore, for small enough \(\varepsilon <0\) , we have that
\(M(\varepsilon ):=s^{-1}(-\infty ,\varepsilon )\subset M \cup U\) 
is an open set of \(N\)  that contains \(\bar{M}\) , and whose boundary
\(\partial M(\varepsilon )=\Omega (\varepsilon )\) 
is a smooth strictly convex hyper-surface of \(N\) . We choose \(\varepsilon \in (0,\varepsilon _0)\)  and set \(\hat{M}=M(\varepsilon )\) .
Let \(p,q \in \bar{M}\)  and choose a distance minimizing unit speed geodesic \(\gamma \colon [0,d_{\hat{M}}(p,q)] \rightarrow \hat{M}\)  that connects these points. Now without loss of generality we may assume that \(\gamma (\tilde{t}) \in U\)  for some \(\tilde{t} \in [0,d_{\hat{M}}(p,q)]\) . If this is not true then the trace of \(\gamma \)  is contained in \(M\)  and we are done.
Since \(U\)  is open and \(\gamma (\tilde{t})\in U\)  we can choose an interval \([a,b]\subset [0,d_{\hat{M}}(p,q)]\)  such that \(\gamma ([a,b])\subset U\)  and define a smooth function
\(\tilde{s} \colon [a,b] \rightarrow {\bf R}, \quad \tilde{s}(t):= s(\gamma (t)).\) 
Since \(p\)  and \(q\)  are in \(\bar{M}\)  we may without loss of generality assume that
\(\tilde{s}(a),\tilde{s}(b)\le 0.\)
We aim to verify that \(\tilde{s}\)  is always non-positive. To establish this we show that the maximum value \(m\in {\bf R}\)  of \(\tilde{s}\)  is attained at the endpoints of the domain interval. So suppose that \(m=\tilde{s}(t_0)\)  is attained in some interior point \(t_0\in (a,b)\) . As \(t_0\)  is a maximum point of \(\tilde{s}\) , laying in the interior of the domain interval, it must hold that \(\dot{\tilde{s}}(t_0)=0\)  and \(\ddot{\tilde{s}}(t_0)\le 0\) . On the other hand since \(\gamma \)  is a geodesic, we have by Weingarten equation [1]} that
\(\dot{\tilde{s}}(t_0)=\langle \operatorname{grad}s(\gamma (t_0)), \dot{\gamma }(t_0)\rangle _g, \: \text{ and } \: \ddot{\tilde{s}}(t_0)=\langle D_t\operatorname{grad}s(\gamma (t_0)), \dot{\gamma }(t_0)\rangle _g=-\Pi _{\gamma (t_0)}(\dot{\gamma }(t_0),\dot{\gamma }(t_0)).\) 
Here \(D_t\)  stands for the covariant differentiation along the curve \(\gamma \) . Therefore \(\dot{\gamma }(t_0)\)  is tangential to the strictly convex hyper-surface \(\Omega (m)\)  which implies that
\(\Pi _{\gamma (t_0)}(\dot{\gamma }(t_0),\dot{\gamma }(t_0))<0.\) 
This in conjunction with (REF ) leads into a contradiction with \(\ddot{\tilde{s}}(t_0)\le 0\) . We have verified that for all \(p,q \in \bar{M}\)  any distance minimizing geodesic in \(\hat{M}\) , between these points, is contained in \(\bar{M}\) . Therefore the equation (REF ) is true.
